How to make Kinetic Falling Grid Effect/animation?

Card Dance - and oldie but a goodie, and a gradient ramp to make the pieces move. 

Start with a comp that is the same size as your grid. Let's say the rectangle is 100 X 100 and Repeater 1 is set to 20 copies 120, 0. Repeater 2 would be 9 copies 0, 120. That means the comp size is 2400 X 1080 and the position of the shape layer would be 60, 60 assuming that the rectangle transform properties were 0, 0. That will give you your grid comp.

Next, create a gradient comp. I used a 50% gray rectangle for the base and 9 copies of a horizontal stroked path + Trim paths and some directional blur to give me an easy way to wipe on a color change row by row. 

The last step is to set up a new comp with the frame rate and frame size you want (Standard HD) add the Gradient Grid comp and the Gradient comp, add Card Dance, and a Comp Camera. You get something like this in just a few minutes:

That old effect by Brian Maffit is probably the best solution for this kind of thing. It's been a while, but I used to use Card Dance all the time. 

If you need extruded shapes like you are showing in your screenshot then C4D (even C4D lite) or Blender is your best option. Trying to work with 180 extruded shape layers in AE and it's pretty easy to get this kind of motion in Blender. 

I bet theres even simpler ways but you can just animate one and duplicate or copy the keyframes to the rest in the grid.  Then offset the keyframes.  There are pay what you want scripts like Rift https://aescripts.com/rift/ and Sortie https://aescripts.com/sortie/  and others to help simplify the process.  Also i always find it easier to work in reverse when i want a bunch of things to come together.  Start with them together then animate it flying out and then reverse the whole thing.